Moduli phenomenology and cosmology Tatsuo Kobayashi 1 Introduction

Moduli phenomenology and cosmology 　　　　　　 Tatsuo Kobayashi 1. Introduction 2. Towards realization of SM 3. Moduli stabilization 4. Moduli/axion inflation 5. Summary

1. Introduction String phenomenology and cosmology 　 Our purpose is to realize the standard model of particle physics 　 to unravel mysteries of particle physics and cosmology from superstring theory. Realization of the SM not just the gauge group, three generations, but quantitative realization such as gauge couplings, Yukawa couplings, etc. 　　　　　

2. Towards realization of SM 10 D superstring theories E 8 x. E 8 Hetero. SO(32) Hetero. Type IIA Type IIB Type I　　　　　　 E-series is the GUT series, E 8, E 7, E 6, E 5(SO(10)), E 4(SU(5)), E 3(SM) E 8 would be interesting. Many people started string pheno. with E 8 x. E 8 hetero.

2 -1. E 8 x. E 8 hetero orbifold models Compactification 10 D 4 Dx 6 D compact space 6 D Calabi-Yau 6 D Toroidal orbifold ……………… We can solve string theory on orbifolds. In principle, We can carry out perturbative calculations. 　　　　 Calculability is not everything, but can tell us something.

Examples of orbifolds S 1/Z 2 Orbifold 　There are two singular points, which are called fixed points.

Orbifolds T 2/Z 3 Orbifold There are three fixed points on Z 3　orbifold (0, 0), (2/3, 1/3), (1/3, 2/3) su(3) root lattice Orbifold = D-dim. Torus /twist Torus = D-dim flat space/ lattice

Closed strings on orbifold Untwisted and twisted strings Twisted strings are associated with fixed points. “Brane-world” terminology: untwisted sector bulk modes twisted sector brane (localized) modes

Explicit Z 6 -II model: Pati-Salam T. K. Raby, Zhang ’ 04 4 D massless spectrum Gauge group Chiral fields Pati-Salam model with 3 generations + extra fields All of extra matter fields can become massive

Pati-Salam ちなみに、Pati-Salam model 10 D N=1 (4 D N=4) E 8 SYM だけでなく、 　　SU(8) SYM, SO(32) SYM にも含まれる

2 -2. Magnetized D-brane models Torus/orbifold with magnetic flux The number of zero-modes, i. e. the generation number, is determined by magnetic flux. Magnetic flux ⇒　non-trival zero-mode profile We know analytical form of profiles ⇒　calculability

Quantum mechanics： Chap. 15 in Landau-Lifshitz book particle in magnetic flux(Landau) Harmonic oscillator at y=k/M M degenerate ground states k=0, 1, 2, …………, (M-1) 　　　　　M=integer

Chiral theory We also apply this for spinor fields. the degenerate (chiral) zero-modes with the same quantum numbers generation numbers of quarks and leptons

Wave functions For the case of M=3 Wave function profile on toroidal background Zero-modes wave functions are quasi-localized far away each other in extra dimensions. Therefore the hierarchirally small Yukawa couplings may be obtained.

U(8) D-brane models Pati-Salam group WLs along a U(1) in U(4) and a U(1) in U(2)R => Standard gauge group up to U(1) factors U(1)Y is a linear combination.

PS => SM Zero modes corresponding to three families of matter fields remain after introducing WLs, but their profiles split (4, 2, 1) Q L

Symmetry breaking by flux and WL Magnetic flux ⇒　non-trivial profile GUT multiplet unbroken gauge boson flat profile Wilson line breaking Q L Broken gauge boson non-trivial profile Coupling between Q and L through heavy bosons is suppressed Hamada, T. K. , ‘ 12

Orbifold with magnetic flux Orbifold 上でも同様 　　 ゼロモードが、ZNの固有値で分類 　　e. g. Z 2 even, odd 世代数と波動関数の変更 Z 2 H. Abe, Choi, T. K. Ohki, Oikawa, Sumita, Tatsuta, 　‘ 08 Z 2, Z 3, Z 4, Z 6 with discrete Wilson lines T. -h. Abe, Fujimoto, T. K. , Miura, Nishikawa, Sakamoto, Tatsuta, ’ 13 -

Magnetic flux background Similarly, we can study E 8 x. E 8 hetero. on torus/orbifolds with magnetic flux Choi, T. K. , Maruyama, Murata, Nakai, Ohki, Sakai, ‘ 10 SO(32) hetero. on torus/orbifold with magnetic flux Abe, T. K. , Otsuka, Takano, Tatsuishi, ‘ 15

(Semi) realistic spectra Visible sector gauge groups SU(3)x. SU(2)x. U(1)Y its extensions with breaking mechanism 3 chiral generations of quarks and leptons no chiral exotics (no chiral extra matter) vector-like exotics (extra matter) would be OK, because they may have mass terms. Usually, there appear lots of singlets. Hidden sector several types of gauge groups, charged matter, (singlets) at high energy scale Some of them may confine, condensate, get masses. 　　

Compactification for realistic spectra Examples (Our models in these ten years) 　E 8 x. E 8 heterotic string on orbifold T. K. , Raby, Zhang, ‘ 04, ’ 05 SUSY Pati-Salam model with 3 generations on asymmetric orbifold Beye, T. K. , Kuwakino, ‘ 14 SUSY SM with 3 generations SO(32) heterotic string on torus with magnetic fluxes 　 (SUSY) SM Abe, T. K. , Ohtsuka, Takano, Tatsuishi, ’ 15 type IIB magnetized D-brane models on orbifold SUSY SM with 3 generations Abe, Choi, T. K. , Ohki, Oikawa, Sumita, Tatsuta, ‘ 09 type IIA intersecting D-brane models on torus SM with 3 generations Hamada, T. K. , Uemura, ‘ 14

Visible sector Other groups also have constructed other (semi)realistic models There are lots of (semi)realistic models in the market for the gauge symmetry and matter content. I cannot count. Maybe more than O(10, 000 -1, 000). Next issue to study is qualitative realization of the SM, values of the gauge couplings, Yukawa couplings, Higgs potential, CP, etc. What about the sector other than the SM, i. e. the hidden sector 　　

2 -3. 4 D effective theory Higher dimensional Lagrangian (e. g. 10 D) integrate the compact space ⇒　4 D theory Coupling is obtained by the overlap integral of wavefunctions

Yukawa couplings 　　　 The Yukawa couplings are obtained by overlap integral of their zero-mode w. f. ’s. ⇒　O(1) coupling suppressed coupling

Stringy computation on Yukawa couplings among localized modes localized far away　　 weak coupling nearly localized moded 　　　 strong coupling heterotic orbifold, intersecting/magnetized u(1) su(2) models su(3) H Q stringy calculation L Y~exp(-area) u, d

Quark/lepton masses and mixing angles Abe, T. K. , Sumita, Tatsuta, ‘ 14 Example of U(8) D-brane models Parameters two moduli ⇒　overall coupling, ratios of Y’s Higgs 、　　２（up） 3（down） from 5 pair higgs Flavor is still a challenging issue.

Quark/lepton masses and mixing angles Abe, T. K. , Otsuka, Takano, Tatsuishi, ‘ 16 Example from SO(32) heterotic string thoery Similar results Flavor is still a challenging issue.

3. Moduli stabilization 10 D 　⇒ 4 D our space-time + 6 D space 10 D tensor 4 D tensor + 4 D vector + 4 D scalars moduli Moduli: geometrical aspects Couplings depend on their VEVs. Moduli is a characteristic feature in superstring theory on compact space.

Couplings Gauge couplings in 4 D depend on volume of compact space　as well as dilaton. Other couplings like Yukawa couplings also Depend on moduli. We need proper values of moduli in order to realize experimental values of gauge and Yukawa couplings.

Moduli Perturbatively flat potential They should have potential, otherwise there are massless scalar fields. Non-perturbative effects generate potential. 　 particle physics and cosmology 　　　low scale SUSY, Inflation, etc Those provide us with characteristic features of superstring on a compact space. kind of prediction

3 -1. Moduli stabilization Our scenario is based on 4 D N=1 supergravity, which could be derived from type IIB string. Flux compactification Giddings, Kachru, Polchinski, ‘ 01 The dilaton S and complex structure moduli U are assumed to be stabilized by the flux-induced superpotential That implies that S and U can have heavy masses of O(Mp). The Kaher moduli T remain not stabilized.

Non-perturbative effects such as D-brane instanton effects 　and gaugino condensation moduli-depnedent superpotential terms D-brane instanton gaugino condensation 　　　　　

Gaugino condensation in hidden sector Strong coupling dynamics 　　 　gaugino condensation b is one-loop beta function coefficient, e. g. b=3 N for SU(N) SYM. T corresponds to

KKLT V Kachru, Kallosh, Linde, Trivedi, ’ 03 SUSY breaking uplifting by SUSY breaking total potential 　　　　　　　　　T before uplifting SUSY point

Moduli　　particle phenomenology For example、low energy SUSY breaking 　　　　　-> unique spectrum of superpartners KKLT scenario mirage mediation 　　　　　Choi, Falkowski, Nilles, Olchowski, ‘ 05 Te. V-scale mirage mediation Choi, Jeong, T. K. Okumura ‘ 05 T. K. Makino, Okumura, Shimomura, Takahashi, ‘ 12 Hagimoto, T. K. , Makino, Okumura, Shimomura 1509. 05327 Still (Ms=O(1 Te. V))　O(10)-O(1)% fine-tuning Cf. CMSSM O(0. 1)-O(0. 01)% tuning 　

Racetrack potential Two terms due to non-perturbative effects 　　　　　

3 -2. SM + moduli stabilization これまでの研究 　SM visible sector の構築と　moduli stabilization は、 　　　独立に研究していた。 　e. g. Visible sector D 3 -brane にlocalize moduli stabilization dynamics 　　D 7 brane 離れている。　　 Visible sector と　moduli stabilization を同時に １つの具体的な模型で考えてみたい。　　　

SM + moduli stabilization Asymmetric orbifold Bye, T. K. , Kuwakino, ‘ 16 heterotic string on asymmetric orbifold　 left-mover と　right-mover を独立に割る 　　　 （６ D幾何学的描像がない） 　moduli は、dilaton しかない。

SM + moduli stabilization Asymmetric orbifold Bye, T. K. , Kuwakino, ‘ 16 starting point Narain lattice before Z 3 orbifolding SU(4)7 x. U(1) for left (gauge), E 6 for right 具体的なstring massless spectrum 　　MSSM + hidden sector (SU(4) x SU(3)x. SU(3)) SU(4)x. SU(3) gaugino condensation racetrack superpotential dilaton stabilization SU(3) SUSY breaking F-term uplifting 　不満足　(hidden sector matter に適当なmass を仮定) 　　

SM + moduli stabilization このタイプの模型の特徴 Hetero は、visible と　hidden の　 gauge coupling が等しい　　　　　　　SM は、２倍。 gravitino mass 　(for MSSM) 　　 N=3 N=4 N=5 10 Me. V 10 Ge. V 10 Te. V

SM + moduli stabilization Magnetized D-brane models Abe, T. K. , Sumita, Uemura, ’ 17 type IIB magnetized D 9 and D 7 models on Z 2 x Z 2 orbifold 　 D 9, D 7 sector Pati-Salam models MSSM-like models magnetic fluxes moduli-dependent FI-term 　　 moduli の比が固定

SM + moduli stabilization Magnetized D-brane models 　　　　　　　　Abe, T. K. , Sumita, Uemura, ‘ 17 type IIB magnetized D 9 and D 7 models on Z 2 x Z 2 orbifold 　 D 9, D 7 sector Pati-Salam models MSSM-like models Dilaton, complex structure moduli は、flux により 　　　stabilize されていると仮定 　この設定で、 可能なD-brane instanton 効果を 　　具体的に計算 (zero-mode 積分) 　　 KKLT 型のmoduli 固定

D-brane instanton: neutrino mass neutrino new zero-modes appears they couple with neutrinos Neutrino masses are induced. Blumenhage, Cvetic, Kachru, Weigand, ’ 06 Ibanez, Uranga, ‘ 06

SUSY breaking and uplifting Abe, T. K. , Sumita, ‘ 16 type IIB magnetized D 9 and D 7 models on Z 2 x Z 2 orbifold 　 D 9, D 7 sector SUSY breaking sector の具体的構築、 visible sector + moduli stabilization sector + SUSY breaking (F-term uplifting) sector Abe, T. K. , Sumita, Uemura, work in progress

3 -3 Radiative moduli stabilization T. K. , Omoto, Otsuka, Tatsuishi, ‘ 17 We assume that one U remains light, while S, T and the other U’s are stabilized with heavy masses by 3 -form fluxes and non-perturbative effects. The real part can be stabilized at

3 -3 Radiative moduli stabilization T. K. , Omoto, Otsuka, Tatsuishi, ‘ 17 The imaginary part is not stabilized because it does not appear in the potential. The F-term SUSY breaking depends on Im(U). SUSY vacuum and SUSY breaking vacuum are degenerate.

3 -3 Radiative moduli stabilization T. K. , Omoto, Otsuka, Tatsuishi, ‘ 17 F-term is determined by Im(U), which is not stabilized by the tree-level potential. If U couples to the visible sector, its F-term induces the gaugino masses and sfermion masses. These induce one-loop potential The explicit form depends on details of the visible sector.

3 -3 Radiative moduli stabilization T. K. , Omoto, Otsuka, Tatsuishi, ‘ 17 simple illustration one set of gauginos, whose masses are a linear function of F one set of sfermions, whose masses squared are a linear function of F The visible sector can stabilize moduli. SUSY or SUSY breaking vacuum depends on the visible sector.

3 -3 Radiative moduli stabilization T. K. , Omoto, Otsuka, Tatsuishi, ‘ 17 We assume that S and U are stabilized with heavy masses by 3 -form fluxes. Kahler moduli have the no-scale structure We assume that the superpotential is independent of Kahler moudli T. 　 Scalar potential is independent of F-terms of T It independent of T if DIW=0.

3 -3 Radiative moduli stabilization T. K. , Omoto, Otsuka, Tatsuishi, ‘ 17 F-terms of T can break SUSY, but the potential is independent of F-term. SUSY vacuum and SUSY breaking vacuum are degenerate. We consider two corrections. 　No-scale structure is violated. One-loop corrections

3 -3 Radiative moduli stabilization T. K. , Omoto, Otsuka, Tatsuishi, ‘ 17 Illustrating potential The visible sector can stabilize F-term of T. The visible sector can stabilize Re(T), but axionic parts remain massless. SUSY or SUSY breaking vacuum depends on the visible sector.

3 -3 Radiative moduli stabilization T. K. , Omoto, Otsuka, Tatsuishi, ‘ 17 At this stage, axionic parts of T’s remain massless. Non-perturbative effects below this energy scale would generate axion potentials. ⇒　　axion inflation, dark matter, axiverse, etc. some interesting aspects

3 -3 Radiative moduli stabilization T. K. , Omoto, Otsuka, Tatsuishi, ‘ 17 Explicit potential depends on the details of the visible sector. It is interesting to study this scenario explicitly in concrete models.

4. Moduli/axion inflation　　 4 D low-energy effective field theory respects some symmetries of moduli, e. g. geometrical symmetries of compact space, gauge symmetries of n-form gauge fields. Then, their potential is flat. Non-perturbative effects break such symmetries, and generate potential. Some symmetries still remain, e. g. continues shift symmetry discrete one modular symmetry, etc. Moduli would be good candidates for inflaton. 　

A scenario for Inflation in string theory: moduli/axion Moduli axion (imaginary part) has a continues shift symmetry flat potential at tree level non-perturbative effects, gaugino condensation superpotential 　 discrete shift symmetry natural inflation Freese, Frieman, Olinto, ‘ 90

A scenario for Inflation in string theory: modular symmetry Circle compactification with radius R winding number momentum Stringy symmetry moduli Stringy non-perturbative effects (such as world-sheet instanton effects ) would appear, but 4 D low-energy effective field theory may respect the modular symmetry somehow.

A scenario for Inflation in string theory: Modular invariant inflation, T. K. Nitta, Urakawa, ‘ 16 Two field inflations by Re. T and Im T by assuming that the others are heavy. Inflaton trajectory along the edge of fundamental region 　　　　　 ns is small.

Threshold corrections S: dilaton, T: Kahler moduli, U: complex structure moduli Heterotic string theory on orbifold: certain twisted sector 　　 Dixon, Kaplunovsky, Louis, ‘ 91 Type IIA intersecting D-brane models: certain parallel D-brane Lust, Stieberger, ‘ 07 T-dual Type IIB D-brane models D 3/D 7 -branes or D 5/D 9 -branes gaugino condensation

Eta function axion inflation in Type IIB Abe, T. K. , Otsuka, ar. Xiv: 1411. 4768 We assume that all of the moduli are stabilized except the axion = Im(U). 　　 Gaugino condensation the decay constant can be large, f> 1.

Poly-instanton axion inflation T. K. , Uemura, Yamamoto, ar. Xiv: 1705. 04088 Non-perturbative corrections on the gauge kinetic function 　　 gaugino condensation Poly-instanton Blumenhagen, Schmidt-Sommerfeld, ‘ 08

Poly-instanton axion inflation T. K. , Uemura, Yamamoto, ar. Xiv: 1705. 04088 Axion potential 　　 One can realize very flat potential.

Poly-instanton axion inflation T. K. , Uemura, Yamamoto, ar. Xiv: 1705. 04088 Axion potential 　　 One can realize very flat potential.

Inflation scenario in superstring theory In the market, we have lots of inflation models in superstring theory, which are consistent with the current experiments. Future experiments would constrain more, but many would remain. It would be important to examine these stringy inflation models by using other aspects, e. g. thermal history after inflation, consistency with construction of the visible sector, etc. 　　

After inflation We know the couplings of moduli to gauge bosons and matter. Through such couplings, moduli (axion) =inflaton decays into visible and hidden matter. We can compute the reheating temperature of the visible sector and abundance of the hidden sector. The couplings are non-universal between the visible and hidden sectors.

After inflation We know the couplings of moduli to gauge bosons and matter. After inflation, the moduli (axion)=inflaton oscillates. Couplings and their phases vary after the inflation. That would have some effects in the history of the Universe, e. g. Baryogenesis Akita, T. K. , Otsuka, 1702. 01604 　　

Summary We have several types of realistic models from superstring theories through compactifications, SU(3)x. SU(2)x. U(1)Y and three generations of quarks and leptons. Next issues is to realize qualitatively particle physics, e. g. Yukawa couplings. Also it is important to study cosmological aspects.

Summary There are several mechanisms for moduli stabilization and lots of inflation models. The search of realistic spectra and study on the inflation sector and moduli stabilization have been studied separately. It is important to study them in one concrete model explicitly.